Then what rational interest of the State is served by denying ss couples "marriages" and marital rights enjoyed by non-ss couples? Particuarly where, as now, some states accord and recognize such rights while other states do not.

That is the core question. For Scalia, it would be simply that the State has a legitimate interest in enforcing the moral consensus of its populace - the same interest that is advanced by laws prohibiting women from going topless at the beach, for example. To quote from his dissent in Lawrence (discussing sodomy, not marriage):

I turn now to the ground on which the Court squarely rests its holding: the contention that there is no rational basis for the law here under attack. This proposition is so out of accord with our jurisprudence–indeed, with the jurisprudence of any society we know–that it requires little discussion.

The Texas statute undeniably seeks to further the belief of its citizens that certain forms of sexual behavior are “immoral and unacceptable,” Bowers, supra, at 196–the same interest furthered by criminal laws against fornication, bigamy, adultery, adult incest, bestiality, and obscenity. Bowers held that this was a legitimate state interest. The Court today reaches the opposite conclusion. The Texas statute, it says, “furthers no legitimate state interest which can justify its intrusion into the personal and private life of the individual,” ante, at 18 (emphasis addded). The Court embraces instead Justice Stevens’ declaration in his Bowers dissent, that “the fact that the governing majority in a State has traditionally viewed a particular practice as immoral is not a sufficient reason for upholding a law prohibiting the practice,” ante, at 17. This effectively decrees the end of all morals legislation. If, as the Court asserts, the promotion of majoritarian sexual morality is not even a legitimate state interest, none of the above-mentioned laws can survive rational-basis review.

Note that none of this is addressing whether the 'promotion of majoritarian sexual morality' might be limited by the rights of people involved. Scalia agrees at the top of the opinion that if gays have a right to engage in any particular activity, then that can certainly trump a legitimate state interest if the appropriate review criteria are satisfied.